Magnetic nanorods: Genesis, self-organization and applications

Article abstract of DOI:10.1524/zpch.2008.222.2-3.229

Magnetic-field-assisted self-assembly of magnetic dipole moment carrying iron nanoparticles is shown to result in the formation of magnetic and mechanically stiff nanoscale rods. The cooperative behavior of an ensemble of such rods and bundles thereof exhibits self-organized pattern formation on different length scales. Pattern formation on large length scales reveals great similarity with physical systems undergoing spinodal decomposition. Possible applications for dipolar magnetic nanorods in the field of perpendicular storage media are highlighted. We discuss an aerosol-synthesis-route allowing to prepare ferrofluids (FF) with shape-anisotropic particles constituting the magnetic phase immersed in the nonmagnetic carrier fluid. These so-called nanorod FF unveil a two orders of magnitude increase of viscosity enforced by an applied field of 10 mT even at shear rates larger than 10^-2 s. This raises prospects for applications in microfluidics and MEMS. by Oldenbourg Wissenschattsverlag.

Full text of DOI:10.1524/zpch.2008.222.2-3.229

Z. Phys. Chem. 222 (2008) 229–255 / DOI 10.1524/zpch.2008.222.2–3.229
© by Oldenbourg Wissenschaftsverlag, München
Magnetic Nanorods: Genesis, Self-Organization
and Applications
∗
By Rainer Birringer , Helmut Wolf, Christian Lang, Andreas Tschöpe, and
Andreas Michels
Universität des Saarlandes, Technische Physik, Postfach 151150, D-66041 Saarbrücken,
Germany
(Received July 10, 2007; accepted July 17, 2007)
Magnetic Nanostructures / Self-Organization / Magnetoviscosity /
Nanorods
Magnetic-field-assisted self-assembly of magnetic dipole moment carrying iron nanopar-
ticles is shown to result in the formation of magnetic and mechanically stiff nanoscale
rods. The cooperative behavior of an ensemble of such rods and bundles thereof exhibits
self-organized pattern formation on different length scales. Pattern formation on large
length scales reveals great similarity with physical systems undergoing spinodal decom-
position. Possible applications for dipolar magnetic nanorods in the field of perpendicular
storage media are highlighted. We discuss an aerosol-synthesis-route allowing to prepare
ferrofluids (FF) with shape-anisotropic particles constituting the magnetic phase immersed
in the nonmagnetic carrier fluid. These so-called nanorod FF unveil a two orders of
magnitude increase of viscosity enforced by an applied field of 10 mT even at shear rates
larger than 10−2 s. This raises prospects for applications in microfluidics and MEMS.
1. Introduction
Recent progress on magnetism and magnetic materials have made nanostruc-
tures a particularly interesting class of materials for both scientific and nanos-
tructured technological explorations. Studies on subjects such as interlayer
coupling, giant magnetoresistance, colossal and tunnelling magnetoresistance,
exchange bias, half-metallic ferromagnets, spin injection and current-induced
switching have eventually led to the exciting possibility of utilizing electron
spin for information processing or spintronics [1, 2]. The materials used for ei-
ther putting the ideas discussed above into practice but also for assembly of
prototypical devices merely belong to the class of layered materials-thin film
*
Corresponding author. E-mail: birringer@nano.uni-saarland.de
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R. Birringer et al.
and multilayer systems. However, not only nanoscale layered materials mani-
fest fascinating and novel properties, a story of scientific and technological
success is evidenced by the large scale application of nanostructured soft and
hard magnetic materials [3], which belong to the category of nanostructured
bulk materials. Nanoscale particulate composites represent a scenario that cov-
ers a broad diversity of materials from biomaterials to superspin glasses, and
granular magnetic materials serving as model systems for the study of ageing,
rejuvenation and memory phenomena [4]. The fabrication of ordered nanos-
trutures – essentially utilizing dot-like or rod-like nanometer-sized objects as
building units – encompasses a variety of preparation techniques such as dif-
ferent lithography and nanoimprint techniques, copolymer nanolithography, as
well as self-assembled and nanotemplate-assisted growth of nanostructures [5].
Fundamentally, novel and unexpected physical effects will emerge when
the sample size and/or some characteristic length scale of microstructure be-
comes comparable or even smaller than a property-dependent characteristic
length scale such as the carrier mean free path, various magnetic exchange
lengths, or the spin diffusion length [6]. This rationale has been the driving
force for the development of increasingly sophisticated materials, as discussed
above. Dimensional analysis of the well-known micromagnetic free energy
yields two fundamental length scales. The wall-width parameter δ0 = (A/K)1/2
(A being the exchange stiffness, and K the magnetocrystalline anisotropy) that
is varying from a few nm in extremely hard materials to more than 100 nm
in very soft materials. It determines the thickness of magnetic domain walls
and the spatial response of the magnetization to local perturbations. Sec-
ꢀ
ondly, the exchange length lex
which describes the competition between interatomic exchange and magne-
tostatic self-interactions; experimentally found exchange lengths are on the
=
A/(µ0 Ms2) (Ms: saturation magnetization)
order of 10 nm [7]. There are other relevant length scales such as the critical
√
2
s
∼
single-domain size [6] RSD = 36 AK/(µ M ) of a sphere (R (Fe) 6 nm,
=
0
SD
∼
∼
R (Co) 34 nm, R (SmCo ) 764 nm) and domain sizes specifying the
=
=
SD
SD
5
real space scale above which inhomogeneous magnetization states extend
throughout the material. In magnets with random magnetic anisotropy, the
magnetic correlation length L ∝ D−3 yields a measure of how many grains of
size D exhibit strong intergranular correlations [8]. A length scale, particularly
relevant to storage media, specifying the size below which the material ap-
pears no longer ferromagnetic is the superparamagnetic limit characterized by
KV ≈ kBT, where V denotes the particle volume; we note that the observation
of superparamagnetic behaviour depends on the time frame of measurement.
In the superparamagnetic state, thermal fluctuations are sufficiently strong to
overcome an effective energy barrier, basically set by the magnetocrystalline-
and shape anisotropy. Finally, there are two limiting length scales in mag-
netism, the Fermi wavelength reflecting intra-atomic electrostatic interactions,
and the correlation length describing the physical behaviour near critical points
e.g. at the ferro- to paramagnetic phase transition [6].
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Magnetic Nanorods: Genesis, Self-Organization and Applications
231
A recent upsurge in research on magnetic nanostructures puts much empha-
sis on understanding of ordered arrangements or patterned arrays of building
units which are confined in two (rod-like) or three dimensions (dot-like) [5].
Advanced sample growth and patterning techniques allows one to control and
modify shape and size of building blocks and likewise the geometry of pat-
terned arrays of such entities. As a result, highly ordered arrangements of
magnetic building units serve now as candidate systems for studying and
understanding mesoscopic effects generated by the simultaneous manifesta-
tion of geometric confinement, physical proximity and presence of structural
order. Altogether, they govern the interplay between the relevant physical
length scales (as discussed above) and the scales and measures related to
size, shape and degrees of imperfections of the building blocks as well as the
geometry/topology of their spatial arrangements.
In studying arrays of nanometer-sized building units, it has been found that
in particular proximity in many cases interferes with the physical length scale
associated with long-ranged interaction phenomena, in this context represented
by the dipolar interaction which decays ∝ 1/r3. In a simplistic approach, as-
suming building blocks which have single domain character and therefore may
be approximated by magnetic dipoles, the dipolar field generated by an indi-
vidual building unit and its interaction with nearest neighbor dipoles may yield
a rather intuitive picture of the magnetic configuration and some of the proper-
ties of the overall array. However, whenever the distance between the building
units becomes comparable to their size, the simple dipolar approach fails.
A more realistic estimate of the effects of dipolar fields in arrays of nanostruc-
tures requires a minimization of the magnetostatic self-energy, which in turn
yields the stable magnetic configuration of the array. This is a formidable task,
since the local values of the magnetization M(r) and the local demagnetizing
field Hd(r) are to be integrated over all sample volume. Nevertheless, many
new phenomena have been discovered and are essentially related to proximity
effects in patterned arrays of nanostructures, such as enhancement or decrease
of coercivity, changes of the switching field width, shape- and interaction-
induced anisotropies as well as shifts of the spin-wave excitation frequencies
to name just the most prominent effects [5].
In summary, it seems fair to conclude that the dipolar interaction mani-
fests a constitutive force in determining and controlling the properties of
ordered arrays of nanostructures, artificially grown or prepared by utilizing
e.g. lithography-, nanoimprinting- or template-assisted growth techniques. In
what follows, we are going to discuss a complementary phenomenon fo-
cussing on the self-organized pattern formation of magnetic nanoparticles
which serve as building units. Again, the dipolar interaction has been identified
as the force of creation, thus controlling and dominating the morphogenesis.
As discussed above, we will treat the individual nanoparticles in a simplis-
tic manner as magnetic dipoles. In chapter two, their single-particle behavior
is discussed in terms of the Stoner–Wohlfahrt model, moreover, the limiting
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R. Birringer et al.
cases of two interacting dipoles and an ensemble of otherwise unconfined
dipoles are addressed. Chapter three deals with the understanding and mod-
elling of the evolution of regular patterns observed during morphogenesis.
Potential ramifications to nanofabrication in the area of high-density perpen-
dicular storage media are highlighted. In chapter four, we discuss how the
magnetoviscous properties of ferrofluids can be enhanced by tailoring the
shape-anisotropy of the magnetic phase dispersed in a nonmagnetic carrier
fluid.
2. Magnetic properties of nanoparticles
2.1 The Stoner–Wohlfarth model
The Stoner–Wohlfarth (SW) model [9] is one of the simplest approaches for
describing hysteresis effects in magnetism. It can be used to describe the
magnetization curve of a collection of noninteracting identical single domain
particles. In the simplest case, the magnetic energy E of an individual particle
has contributions from uniaxial magnetic anisotropy and from the interaction
of the particles’ magnetization vector M with an applied magnetic field H,
E = KV sin2 θ −µ0VH·M .
(1)
Here, K denotes the effective uniaxial anisotropy constant, V is the volume
of the particle, θ represents the angle between the anisotropy axis and M, and
µ0 is the permeability of free space. Within the SW model, the particles are
assumed to be sufficiently small so that strong exchange forces hold all the
atomic spins parallel and, consequently, any space dependence of the magneti-
zation can be neglected. On the other hand, the particles should be large enough
to avoid temperature effects (see below).
For given materials parameters and for a given orientation of the anisotropy
axis with respect to the applied field, the magnetization curve of a single par-
ticle can be obtained by minimizing Eq. (1) with respect to the angle between
M and H. When the uniaxial anisotropy axis is parallel to H, the hystere-
sis loop has the well-known rectangular shape, with a remanence Mr which
is equal to the saturation magnetization Ms and with a coercivity (switching
field) Hc0 = 2 K/(µ0 Ms). On the other hand, no hysteretic behavior results for
a perpendicular orientation between the anisotropy axis and H. For a sam-
1
ple of randomly oriented noninteracting uniaxial particles, Mr = Ms and
2
0
c
∼
Hc 0.479 H [9].
=
2.2 Magnetic anisotropy
The origin of the assumed uniaxial magnetic anisotropy in Eq. (1) may be
due to the dipole–dipole interaction (shape anisotropy) and/or due to the com-
bined effect of the crystal-field and spin–orbit interaction (magnetocrystalline
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Magnetic Nanorods: Genesis, Self-Organization and Applications
233
anisotropy). Other types of anisotropy which may be of relevance in the context
of nanoparticle magnetism such as surface anisotropy or exchange anisotropy
will be ignored in the following discussion. The general expression for the
magnetostatic shape-anisotropy term EM of a homogeneously magnetized el-
lipsoid is [10]
ꢁ
ꢂ
1
EM = µ0V Nx Mx2 + Ny My2 + Nz Mz2
,
(2)
2
where Nx , Ny, and Nz denote the demagnetizing factors along the principal
(e.g., Cartesian) axes of the ellipsoid, and Mx, My, and Mz are the respective
Cartesian components of the magnetization. For the particular case of an elon-
gated cigar-shaped prolate spheroid (with two equal axes), Eq. (2) simplifies
to
1
EM = µ0V(Nx − Nz)Ms2 sin2 θ +const. ,
(3)
2
where we have assumed Nx = Ny and Mz = Ms cos θ. Equation (3) reveals the
same functional form as the first term on the right hand side of Eq. (1), with the
1
effective shape-anisotropy constant K = Ks = µ0 (Nx − Nz) Ms2.
The magnetocrystalline anisotropy energy2EK expresses the dependency
of the magnetic energy of a particle on the orientation of its magnetization
M relative to the crystal axes. For a particle with uniaxial magnetocrystalline
anisotropy, EK is generally expanded in terms of the angle θ between M and
the anisotropy axis, which is e.g. the c-axis in hcp Co,
EK = K1V sin2 θ + K2V sin4 θ +. . . ,
(4)
where K1 and K2 are functions of temperature. The use of cubic magnetocrys-
talline anisotropy in Eq. (1) instead of or in addition to uniaxial anisotropy as
well as the incorporation of higher-order (K2) terms is straightforward.
It is of interest to compare the magnitude of magnetocrystalline anisotropy
with the strength of shape anisotropy. At room temperature, the magnetocrys-
talline anisotropy constants of single crystals of Ni, Fe, and Co are, respec-
tively, −5.7×103 J/m3, 4.7×104 J/m3, and 4.5×105 J/m3 [11]. By using the
saturation-magnetization values Ms(Ni) = 500 kA/m, Ms(Fe) = 1700 kA/m,
and Ms(Co) = 1400 kA/m, we see that for an elongated particle (with Nx =
1/2 and Nz = 0), the corresponding values for the shape-anisotropy constants,
7.9×104 J/m3, 9.1×105 J/m3, and 6.2×105 J/m3, are (at least for Ni and Fe)
much larger than their magnetocrystalline counterparts.
2.3 Pole avoidance
The dipole–dipole interaction is usually the most complicated energy term,
since it requires evaluating the interaction of each atomic magnetic moment
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R. Birringer et al.
Fig. 1. Bright-field TEM image of an ensemble of Fe nanoparticles. The pole avoidance
principle supports the formation of flux-closure patterns (ring structures).
with the magnetic field which is created by all the other dipoles. The com-
plexity of the problem can easily be grasped by considering the well-known
textbook case of two isolated dipoles. Equation (5) describes the mutual inter-
action energy E1,2 between two magnetic moments m1 and m2 separated by
a distance r1,2 (r1,2 = |r1,2|),
ꢃ
ꢄ
µ0 m1 ·m2 3 (m1 ·r1,2) (m1,2 ·r1,2)
E1,2
=
−
.
(5)
4π
r13,2
r15,2
When the dipoles are forced to align in parallel with the moment direction
along the separation distance, then the parallel “head-to-tail” configuration is
the one with the lowest energy. On the other hand, when the collinear dipoles
are constrained to have their moments oriented perpendicularly to the separa-
tion distance, antiparallel alignment is energetically favoured. This relatively
simple example suggests that in realistic experimental situations, where usually
a large number of particles, wires, etc. are involved, the problem of finding the
ground state of the magnetostatic energy is rather complicated, and it is difficult
to predict whether the effect of the dipolar interaction is e.g. to prefer parallel
or antiparallel alignment of magnetic moments.
Nevertheless, based on the energetics, some general statements on the qual-
itative behaviour of a large number of building blocks can be made [10].
The magnetostatic energy is positive definite, EM ≥ 0, and prefers magneti-
zation structures with no so-called surface and volume poles, i.e., configura-
tions where both terms n·M (surface charges) and −∇ ·M (volume charges)
vanish; here, n denotes a unit vector which is normal to the surface of the
magnetic specimen. The above statement is known as the pole-avoidance prin-
ciple. The most prominent manisfestation of this rationale is the formation of
so-called flux-closure patterns which are characterized by the tendency to min-
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Magnetic Nanorods: Genesis, Self-Organization and Applications
235
imize EM. As an example, Fig. 1 shows the realization of pole avoidance on
the nanoscale: eventually, a chain of sufficiently large nanoparticles will bend
and form a closed ring structure, once the number of particles reaches a certain
critical value.
2.4 Magnetic relaxation
As indicated in Sect. 1, the SW model is valid only at zero temperature. At
nonzero temperatures, the thermal energy kBT, where kB denotes the Boltz-
mann constant, may induce transitions between local energy minima. When
the associated relaxation time is comparable to the characteristic time of the
measurement, which in the case of dc magnetization is typically a few sec-
onds, time-dependent effects become important, since the magnetic properties
change during measurement.
Within the Ne´el–Brown theory of superparamagnetism, the magnetic re-
laxation time τN of a particle exhibiting uniaxial magnetic anisotropy is given
by [10]
ꢅ
ꢆ
KV
τN = τ0 exp
,
(6)
kBT
where τ0 10 −10−10 s, and KV denotes the anisotropy-energy barrier. At
−9
∼
=
zero applied magnetic field, Eq. (6) describes the average time it takes for the
magnetic moment to jump between the two equivalent energy minima at θ = 0
and θ = π (compare Eq. (1)).
When the particles are dispersed in a fluid of viscosity η, an additional re-
laxation mechanism, the so-called Brownian rotational-diffusion mechanism
comes into play. Brownian relaxation is characterized by a switching time
3 VH η
τB =
,
(7)
kB T
where VH denotes the hydrodynamic volume of a single particle. In general,
both mechanisms Ne´el and Brown relaxation take place, resulting in an effect-
ive relaxation time [12]
τN τB
τeff
=
.
(8)
τN +τB
3. Self-organized pattern formation
of magnetic nanoparticles
3.1 Self-organization
Ordinary states of matter like a layer of fluid, a mixture of chemicals, gran-
ular or genetic material can exhibit under appropriate conditions a multitude
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236
R. Birringer et al.
of self-organization phenomena on a macroscopic scale in the form of spa-
tial patterns, temporal rhythms or spatiotemporal order [13]. We may regard
such patterns as global emergent properties of the system, since the scale of
ordering vastly exceeds the range of interactions of the constituents, and there
is no obvious hint of a length scale of this magnitude in the microscopic
physics of the unpatterned state. In its essence, the understanding of pattern
formation relies on the concept of instabilities of a reference state [14, 15].
In experiment, such instabilities arise when a system in equilibrium is driven
into non-equilibrium states by increasing an externally manipulable control pa-
rameter. Along its path the dynamical system eventually becomes infinitely
sensitive to fluctuations (perturbations) and responds by growing disturbances
– the onset of symmetry breaking – in a spatially uniform state [16]. The cru-
cial next step in the pattern-forming process rests on some built-in nonlinear
mechanisms that enable the driven away-from-equilibrium system to create and
sustain new states of matter, displaying regulatory behavior in space and/or
time which would be exceedingly improbable to occur under equilibrium con-
ditions [14, 17]. Non-equilibrium spontaneous pattern formation implies that
the constituents of the system must be able to communicate via an extraordi-
nary manner of cooperativity over distances much longer than those to which
they are accustomed at equilibrium.
3.2 Experimental
We study magnetic-field-assisted self-assembly of magnetic dipole moment
carrying aerosol-grown iron nanoparticles in an aerosol flow condenser. The
precursor iron pentacarbonyle is evaporated at a rate of 1 µl/min into a con-
tinuous flow of 500 standard cubic centimeter per minute (SCCM) of inert gas
argon at a pressure of 15 mbar and thermally dissociated in a furnace held at
950 ◦C. The released iron atoms grow by condensation and coagulation and fi-
nally form magnetic nanoparticles. Further downstream they are collected on
a liquid nitrogen cooled flat substrate in the presence of a homogeneous mag-
netic field of 0.2 T, oriented perpendicular to the substrate plane; the details
of the preparation are discussed in [18]. Enforced by the dipolar interaction,
the early stage of particle collection is dominated by a head-to-tail arrange-
ment of individual particles. Assisted by the external field, the system further
evolves by forming arrays of string-like objects. The homogeneous field is ap-
plied to prevent the system from taking a path into a magnetically disordered
(frustrated) state, which is difficult or even impossible to analyze. The task
of the applied external field is twofold: first, it aligns the particles moment
head-to-tail perpendicular to the substrate plane, and secondly, the flux of the
overall field becomes concentrated along the already deposited particles re-
sulting in localized field gradients which exert attractive forces on arriving
particles thereby guiding them to the head-ends of the already deposited par-
ticles. This process is self-sustaining and strongly favors parallel arrangement
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Magnetic Nanorods: Genesis, Self-Organization and Applications
237
of individual dipoles and, likewise, supports the formation of string-like par-
ticle chains. Therefore, what actually remains to be studied and understood is
the collective behavior of such string-like magnetic moment carrying objects in
the presence of a homogenous i.e. no force exerting magnetic field of variable
strength.
Evidence for the formation of particle chain segments, driven by dipolar
interaction, has been found in ferrofluids (dispersions of magnetic colloids)
even in zero magnetic field [19]. When applying an external field, the dipo-
lar character of the linear chain segments leads to their alignment along the
field direction and a concomitant transition from individual chains to large ag-
gregates of chains. Chain–chain interaction between nearly parallel rod-like
dipolar objects has been proposed to result in attraction between parallel ori-
ented chains [20]. However, the experimental study of chain–chain interaction
in dipolar fluids is hampered by the fact that particle chains are fluctuating
objects that thermally dissociate (chains “melt” with increasing temperature)
and again recombine to chains with decreasing temperature [21]. Moreover, the
spatial and orientational correlations of individual chains crucially depend on
the magnetic coupling between individual particles [22], which is strongly in-
fluenced by surfactants covering the particles’ surfaces. As a result, it seems
that a necessary prerequisite for studying and understanding the collective be-
havior of one-dimensional dipolar objects is solid-like rigidity between neigh-
boring particles, so that string-like objects can be treated as mechanically stiff
magnetic rods.
In the following, we are going to demonstrate that aerosol-grown magnetic
iron nanoparticles when deposited on a substrate in the presence of a homo-
geneous magnetic field form an array of dipolar rigid rods. It is the reduction
of the particles’ free surface energy that yields the driving force for neck
formation (sintering) [23] between neighboring particles, thus favoring the
emergence of solid magnetic rod-like entities. With increasing particle sup-
ply, the rod length increases, bundles of rods appear, and eventually the system
undergoes a hierarchy of structural instabilities to form self-organized pat-
terns. A schematic of successive stages of morphogenesis is shown in Fig. 2.
The experimentally observed structures and patterns are displayed in Fig. 3.
Obviously, self-assembled structures occur on all length scales starting at the
nanometer scale but also covering the micrometer regime. We consider the hex-
agonal pore carpet as the most prominent feature of pattern formation and,
therefore, we direct our focus on understanding the emergence of pore pattern
in the following paragraphs.
3.3 Pattern formation: modelling and discussion
In order to rationalize the process of pore formation, we assume bundle density
fluctuations to appear in the homogeneous bundle carpet by laterally bending
of individual bundles. At short bundle lengths, the solid-like rigidity of individ-
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238
R. Birringer et al.
Fig. 2. Schematic representation of the evolution of form and patterns during magnetic
field assisted deposition of iron nanoparticles on a non-magnetic substrate. (a) The parti-
cles’ magnetic moment (white arrows) becomes aligned parallel to the external field H0
during deposition. (b) Magnetic field assisted self-assembly of individual dipolar rods (for
more details see Sect. 3.3). The build-up of field gradients is symbolized by flux-lines.
(c) Further growth of individual rods is accompanied by a change in growth mode re-
sulting in the formation of bundles of rods. Each bundle having a typical diameter of
1 µm constitutes a network of a few hundred rods. The center-to-center distance d0 be-
tween nearest neighbor bundles is about 5 µm. (d) The system of bundles carrying the
magnetic flux gives now the appearance of a carpet with homogeneous bundle density
n0 ∝ 1/d02, and bundle length ꢀ. The consequences of bundle–bundle interactions are sum-
marized in (e). (e) When the carpet grows beyond a critical length of ꢀc ≈ 40 µm, the
bundles start bending so that funnel-like pores of zero-bundle-density with a typical dis-
tance of λc ≈ 300 µm develop. Since the number of bundles is conserved, pore formation
is associated with an enhancement of bundle density in the space around pores.
ual bundles gives rise to an elastic restoring force, which limits the amplitude
of fluctuations to infinitesimally small values. With increasing bundle length ꢀ,
the system of homogeneous bundle density is driven into non-equilibrium and
becomes sensitive to infinitesimal density fluctuations. Eventually, symmetry
becomes broken by spontaneously developing pores. It appears that individual
pores emerge by tilting bundles radially outward relative to the pore-center axis
thereby generating bundle density gradients in the space between pores. We de-
scribe the geometry of a pore by introducing the tilt angle a(r, t) that measures
the obliquity of bundles relative to its pore-center axis. The lateral arrangement
of pores across the substrate exhibits hexagonal symmetry implying that the
relevant constituents of the system must be able to communicate in a highly
cooperative manner over distances which are comparable to the system size of
tens of millimeters.
The fundamental observation, also manifesting a mode of collective be-
havior, concerns the tilt angle a: tilt angles of individual pore forming bundles
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Magnetic Nanorods: Genesis, Self-Organization and Applications
239
Fig. 3. Pattern formation in a 40 µm thick layer of iron aerosol particles deposited in a ho-
mogeneous magnetic field. (a) TEM bright-field micrograph of an isolated Fe-particle
chain, obtained by ultrasonic disintegration of a Fe-chain bundle. (b) TEM bright-field
micrograph of a single Fe-chain bundle, grown on a TEM grid at a low deposition rate,
illustrating entanglement of individual particle chains at large ꢀ. (c) Top-view SEM mi-
crograph of a 40 µm thick layer of Fe-chain bundles manifesting self-organized density
modulations in the form of blossom-like pores arranged in a pattern exhibiting (local) hex-
agonal symmetry. The core regions of edge dislocations in the ordered arrangement of
pores are marked by circles. (d) Magnified single blossom-like pore of (c). All SEM and
TEM viewgraphs were taken ex situ. Exposure of the as prepared structures and patterns
to ambient atmosphere resulted in thin oxide layer protecting the Fe-core structures [24].
cancel out when averaged over the total substrate area A so causing aꢀA = 0. In
other words, the tilt angle is a globally conserved quantity and, therefore, must
satisfy a continuity equation
∂
aꢀA = 0
⇒
a = −∇ ·J ,
(9)
∂t
where J is interpreted as a tilt-angle current in full analogy to diffusive flux of
matter in a system of conserved particle number. In such systems the current
is proportional to the local chemical potential gradient and the chemical poten-
tial itself is given by the change of free energy density with respect to particle
density. Following this guideline, we assume a linear response ansatz for J
ꢅ
ꢆ
δw
J = −M∇
,
(10)
δa
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240
R. Birringer et al.
here M describes the mobility of the bundles’ top ends when bending occurs,
and w is an appropriate energy functional density.
The experimental evidence aꢀA = 0 together with the fundamental Eqs. (9)
and (10) are the basic ingredients we will use to model and understand pore-
pattern formation. Combining Eqs. (9) and (10) yields the nonlinear partial
2
differential equation of the problem ∂a/∂t = +M∇ (δw[a]/δa), the solution
of which is a formidable task. We prefer simplifying this equation based on
assumptions motivated by experimental observations. As we observe bundles
being always tilted towards increasing bundle density and never orthogonal to
the density gradients, likewise, any chiral character of bundle arrangement is
missing, it seems feasible to describe structure formation by a one dimensional
approach. Beyond the critical length where the symmetry of the homogeneous
system becomes broken, it appears appropriate to orient our coordinate axis to
be parallel to the substrate and parallel to the wave vector k representing the ob-
served density modulation of the pattern. The remaining task is to construct an
expression for the total energy density w.
We consider three relevant energy terms entering w. Modelling individ-
ual bundles by magnetic dipolar rods, the magnetic dipolar interaction energy
density wm is an obvious contribution to w. Taking only nearest-neighbor
interactions into account, wm is obtained by a Taylor expansion about the ho-
mogeneous bundle configuration a ≡ 0, and constitutes a nonlinear functional
of a(x) and its gradients. The energy associated with bending of bundles with
solid-like rigidity is modelled by the bending energy density of a single solid
beam, wel, multiplied by the local bundle density n(x) = n0 + δn(x) = n0 −
n0ꢀ (∂xa) with n0 being the bundle density of the homogeneous carpet; overall,
we obtain wel ∝ a2/ꢀ. Similarly, the interaction energy density wh with the ex-
ternal field H0 is modelled by that of a single dipole with H0 multiplied by the
local bundle density n(x); wh = −µ0 n(x) mH0 cos[a(x)] where m is approxi-
mated by the saturation magnetization Ms of a single rod (m ∝ Msꢀ). Finally,
we obtain for the total energy density
w[a(x)] = c0 +c2a2 +c4a4 +η(∂xa)+κ(∂xa)2 +· · · ,
(11)
where c0, c2, c4, η, κ are coefficients representing the individual energy densi-
ties or combinations thereof.
The one dimensional version of the combined Eqs. (9) and (10) inserted
with the expression for w reads after linearization
∂
d2a
dx2
d4a
dx4
a(x, t) = (c2 M)
−(2κM)
.
(12)
∂t
Equation (12) reflects full analogy to the linearized Cahn–Hilliard equation
(C–H eq.) of spinodal decomposition [25–27]. The coefficient
ED4
40ꢀd02
3µ0 m2 µ0 mH0
c2 =
−
+
(13)
d05
2d02
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Magnetic Nanorods: Genesis, Self-Organization and Applications
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is positive for short rod length, thus, being dominated in the early growth stage
by mechanical stiffness of the rods (wel). However, finally c2 becomes nega-
tive due to the power law increase of the dipolar interaction term (m2 ∝ ꢀ2)
with increasing bundle length. The coefficient κ remains positive at all length
scales. As a result, our analysis clearly identifies ꢀ as the external control
parameter, and the critical length ꢀc marking the onset of instability is de-
termined by c2(ꢀc) = 0. The finding that the external field contribution be-
comes eventually dominated by the dipolar interaction term justifies our notion
of magnetic-dipole-interaction-driven spinodal decomposition as a plausible
mechanism which triggers spontaneous growth of long-wavelength perturba-
tions in a spatially uniform state. In its essence, the emergence of pores is
caused by repulsive dipolar forces; attractive dipolar interactions are effec-
tively suppressed by the missing lateral and horizontal degrees of freedom of
movement of individual bundles relative to the substrate plane. Nevertheless,
the externally applied field is a necessary prerequisite for pattern formation
since, first, H0 forces the magnetization of the incoming particles to orient par-
allel to it, and second, the interaction of already deposited particles with H0
generates the field gradients which stimulate the growth of linear chains and
bundles.
We conclude by discussing the central quantitative results obtained from
the C–H eq. (12) which is solved by
a(x, t) = ak exp{A(k) t} exp(ikx) ,
(14)
with A(k) = −M(c2 k2 +2κ k4), and k denotes the wave number of a bundle-
density-modulation wave. The solution consists of a spatially periodic term
and a term describing the time evolution of the amplitude. If A(k) < 0 ∀k,
all modes k will decay. For ꢀ > ꢀc, modes exist for which A(k) > 0, con-
sequently, the amplitudes grow exponentially. We assume that the fastest
growing mode kc, determined by dA/dk = 0, selects the characteristic length
λc = 2π/kc of the evolving pattern. We find λc ≈ 1000 µm and ꢀc ≈ 47 µm
which agree, despite the restrictive assumptions, fairly good with the ex-
perimentally found values: λecx ≈ 300 µm and ꢀecx ≈ 40 µm. Strong evidence
for the validity of our simplistic analysis of pore-pattern formation has been
obtained by experimentally studying the external field dependence of λc.
Based on the condition dA/dk = 0, theory predicts kc2 = |c2|/(4κ) result-
ing in λc ∝ H−1/2– if H0 is not too small. This power law dependence of
0
λc as a function of the externally applied magnetic field H0 has been rea-
sonably well verified by experiment, see Fig. 4. The observed influence of
the field confirms that the pattern formation occurs during deposition and
not during removal of the applied field at the end of the deposition pro-
cess. Finally, we notice that the observed non-equilibrium pore-pattern for-
mation belongs to a subclass of dynamical systems, namely, gradient or
potential systems characterized by ∂a/∂t = δF[a]/δa [14]. For such sys-
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242
R. Birringer et al.
Fig. 4. Scaling behavior of critical density modulation wavelength λc. Normalized average
nearest neighbor distance λc/ꢀc of blossom-like pores plotted as a function of the normal-
ized applied magnetic field µ0 H0/n0Φ at fixed bundle length ꢀc on a log–log-scale (n0:
homogeneous bundle density; µ0: permeability constant; Φ: magnetic flux through a sin-
gle bundle). The solid line represents a least-squares fit to the data points. From the slope,
0.06
we determine the power law λc ∝ H0−0.54
.
tems the dynamics consists of relaxation towards the minimum in the en-
ergy functional F in agreement with the behavior predicted by the C–H
equation.
3.4 Possible applications
Apart from pattern formation, magnetic field directed self-assembly may have
potential ramifications to nanofabrication, especially in the area of high-density
perpendicular magnetic storage media [28–30]. We suggest a manufacturing
process that combines lithographically defined generation of ordered struc-
tures, which can be created over large areas, with magnetic field directed
self-assembly of nanometer-sized static magnetic moment carrying aerosol
particles [31]. In fact, our idea relies upon the manufacturing of a lithograph-
ically defined master substrate encoding the desired width of and distance
between magnetic tracks, as well as enabling to adapt rotationally symmetric
arrangement of tracks – the standard read and write geometry. As shown in the
previous paragraphs, magnetic field assisted deposition of magnetic nanopar-
ticles on such a substrate would result in rod-like particle arrangements deco-
rating with high preference the lithographically defined magnetic tracks of the
master substrate. Diameter of rods, and shape anisotropy – which may become
much larger than the intrinsic anisotropies particularly in nanostrucutres –, as
well as material selection would allow to tailor the key parameter of magnetic
recording media [5, 32, 33]. Figure 5 illustrates the two-step process suggested
to prepare nanoscale perpendicular magnetic storage media.
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Magnetic Nanorods: Genesis, Self-Organization and Applications
243
Fig. 5. Manufacturing process of a perpendicular magnetic storage medium by magnetic
field directed self-assembly. (a) Schematic topview of a perpendicular magnetic storage
medium master substrate which is a prerequisite for high throughput mass production.
A cross-sectional view of the marked rectangular frame in (a) is shown in (b). (b) Litho-
graphically generated high permeability (high µr) tracks. The buffer layer enables to lift
off the deposit from the master substrate. In what follows, we describe the steps necessary
to build-up the magnetic storage medium: Step 1 (c), magnetic field assisted deposition of
magnetic aerosol particles forming nanorods by sintering. Step 2 (d), mechanical stabilisa-
tion and chemical passivation by a filler material. Step 3 (e), attachment of a carrier sub-
strate on the magnetic storage layer. Step 4 (f), perpendicular magnetic storage medium
lifted off from the master substrate.
4. Ferrofluids (FF)
Interestingly, the impact of tailor-made magnetic nanoparticles has not only in-
fluenced the development of novel solid state nanostructures and devices but
also the realm of complex fluids has benefitted from the availability of an
ever growing variety of magnetic nanoparticles. Ferrofluids (FF) [34] mani-
fest a scientifically and commercially relevant example of a smart complex
fluid, consisting of magnetic nanoparticles, typically spherically-shaped, with
diameters on the order of 10 nm and with adsorbed dispersant layers of ap-
proximately 1–2 nm thickness, enabling to stably immerse such particles in
nonmagnetic carrier fluids. They decorate magnetic field gradients, which gave
raise to a number of commercially viable applications of FF such as rotary
seals, stepper motor dampers or heat transfer fluids in audio speakers. Owing
to the possibilities for designing and modifying the physical and chemical
properties of both the suspending fluid and the particles as well as the ad-
sorbed dispersant layer, FF are gaining popularity in several novel applications,
functioning as magneto-responsive colloidal extractants, targeted magnetic car-
riers, in vivo imaging agents and in magnetic cell-sorting schemes [35]. Those
nanoparticles represent either permanent magnets, or behave as superpara-
magnetic particles. Suspended in carrier fluids, they undergo rotational and
translational Brownian motion. The free rotation of magnetically hard par-
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244
R. Birringer et al.
ticles in a carrier liquid exhibits a magnetization versus field behavior re-
sembling that of superparamagnetic particles, yet is of different origin. It is
the fascinating interplay between magnetic and rheological properties that
has generated much cross-disciplinary interest in the nature and behavior of
FF [36].
A central issue of fundamental research on FF is magnetoviscosity, dis-
covered by McTague and Rosensweig et al. [37, 38], delineating that in the
presence of a homogeneous magnetic field the magnetic torque on magnet-
ically hard particles hampers or even prevents them from free rotation in
a shear flow as dictated by the fluid dynamics of the carrier liquid. By tak-
ing into account the magnetic and mechanical torques as well as the ther-
mally induced Brownian motion, Shliomis [39] delivered the first theoret-
ical description of the magnetoviscous effect for a FF in the dilute regime
of magnetically non-interacting, spherical particles. As a result, the viscos-
ity rises with increasing magnetic field H and the relative viscosity ∆η/η0
reaches a plateau-value of 1.5 φ, where η0 = η(H = 0), and φ is the nanopar-
ticle volume fraction. Recently, Odenbach and coworkers [40] investigated
the magnetoviscous behavior of highly concentrated (φ ꢀ 0.1), commercially
available FF and found changes of viscosity more than a hundred times
larger than the values expected from theory [41]. It has been verified that
this dramatic increase of viscosity is due to dipolar-interaction-driven for-
mation of chain-like aggregates of individual particles. A second observa-
tion manifests that with increasing shear rates, γ ꢀ 10 s−1, the giant vis-
˙
cosity enhancement disappears, which has been attributed to the fragile na-
ture of the dipolar coupled chain-like aggregates; in fact, it has been sug-
gested that they become disintegrated by shear forces supplied by the carrier
fluid [42].
Irrespective of fundamental research issues, the envisioned novel applica-
tions or the diversity of commercially viable applications of FF, the currently
used FF manifest a common characteristic: namely, the spherical/equiaxed
shape of the constituent magnetic particles. Regarding their magnetic proper-
ties, the effect of strength of the magnetocrystalline anisotropy energy on the
magnetoviscosity has been theoretically investigated quite recently [43]. We
propose to explore a plausible, yet heretofore scarcely noticed degree of free-
dom: the shape-anisotropy of the constituent particles, which can be tailored
to enhance various FF-properties. In particular, we will prepare and charac-
terize rod-like magnetic nanoparticles, which behave as Brownian particles
when immersed in a carrier fluid, therefore, named: nanorod-FF. The rod-like
instead of sphere-like geometry of the magnetic phase, first, suppresses super-
paramagnetism as well as typically overcomes the crystalline anisotropy of the
particles due to the large shape anisotropy energy and, secondly, anisometric
permanent magnets with solid-like rigidity are expected to resist shear flow
and hence contribute to magnetoviscosity far more effectively than equiaxed
particles.
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Magnetic Nanorods: Genesis, Self-Organization and Applications
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4.1 Nanorod ferrofluids: basic ideas
Based on a Fokker–Planck equation, Martsenyuk, Raikher and Shliomis [44]
derived an expression for the magnetic-field-dependent change of viscosity,
∆η, of a dispersion of non-interacting nanoparticles exposed to homogeneous
shear flow, given as:
1
∆η = φ τ M(B) B .
(15)
⊥
4
The applied magnetic field is labeled by B, the transversal relaxation time τ
⊥
translates into the Brownian relaxation time τB via τ = 2τB/(2 + ξL(ξ))
⊥
where L denotes the Langevin function L(ξ) = coth(ξ)−1/ξ with its argument
ξ = mB/kBT, and m stands for the magnetic moment of an individual particle.
M(B) indicates the field-dependent magnetization of the FF which for small
shear flow according to [44] may be approximated by M(B) = Ms L(ξ) and
Ms = mtot/Vmag is the saturation magnetization of the sample. The overall num-
ber of particles making up the magnetic volume Vmag and occupying the volume
fraction φ of the FF have the overall magnetic moment mtot. It seems instructive
to consider the limiting case of saturation, where a sufficiently large applied
field, Bsat, assures that the Langevin function approaches a quasi-plateau-value
characterized by the saturation magnetization Ms. It is straightforward to derive
that by approaching saturation at fixed temperature T, say room temperature,
Eq. (15) simplifies to
∆η(Bsat) ∝ φ τB .
(16)
The particle geometry is hidden in τB = (2Dn)−1, with Dn being the coefficient
of rotational diffusion. As shown by Brenner [45, 46], Dn of solids of revolu-
tion – here we consider long slender axisymmetric bodies, e.g. cylinders – can
be written as:
1
K(n)
=
,
(17)
Dn
Ds
where K(n) is a shape function, n specifies the aspect ratio of bodies and
Ds = kB T/6 η Vhsyd is the Stokes rotational diffusion coefficient of a sphere with
equal hydrodynamic volume, Vhsyd ≡ Vhcyydl. Mathematical expressions for shape
functions are given in references [45–49]. For the sake of argument, it seems
sufficient to display the functional form of K(n) as shown in Fig. (6) for prolate
ellipsoids and circular cylinders [45, 46], where
ꢃ
ꢄ
2
9
n2
n2
(ln n)2
3 L
8π
K(n) =
+
(ln 2−1)+
(18)
ln n
denotes the shape function of a blunt-ended cylinder with aspect ratio n. The
numerical constant L depends critically on the precise shape of the blunt ends
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246
R. Birringer et al.
Fig. 6. Shape function K(n). Functional behavior of K(n) versus aspect ratio n for cylin-
ders (dashed line) and prolate ellipsoids (full line).
of the body; L ≈ 5.45 has been determined for a circular cylinder. Requir-
ing Vhsyd ≡ Vhcyydl(n) and substituting Eq. (18) into Eq. (17) yields an expression
for the rotational diffusion coefficient and, hence, also for τB which depends
apart from temperature on the material parameters η, Ms, φ and n. Let us con-
sider spheres (n = 1) as reference system, and further assume that all material
parameters except n are fixed at constant temperature. Then, increasing n at
conserved Vhsyd consequently contributes with a nonlinear enhancement of τB
and thus ∆η due to the effect of shape anisotropy on Dn imparted by K(n). We
note that τB = (1/2π) fB−1 is inversely proportional to the Brownian relaxation
frequency fB, which is quantitatively accessible by means of ac-susceptibility
measurements. This allows one to deduce an overall aspect-ratio of anisomet-
ric objects, probed in the state of being immersed in a carrier fluid, based on the
relation
1
πK(n)
=
,
(19)
fB(n)
Ds
provided that all other material parameters are prescribed. Finally, since fB(n)
renders τ becoming n-dependent, the magnetic-field-dependent viscosity of
⊥
nanorod FF [Eq. (15)] is suggested to crucially rely upon n, which, when ex-
ceeding 10, may give rise to a giant magnetoviscous effect. We will scrutinize
this expectation by performing field and frequency dependent viscosity meas-
urements on a nanorod FF in a squeeze flow rheometer.
4.2 Synthesis and characterization
Iron nanorods were generated in a modified aerosol flow condenser [50] as
shown in Fig. 7. Before evaporation, the reactor was evacuated to 10−6 mbar.
Then the high vacuum pump was disconnected and replaced by a rotary vane
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Magnetic Nanorods: Genesis, Self-Organization and Applications
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Fig. 7. Magnetic-field-assisted aerosol synthesis of Fe-nanorods. Iron particles are grown
in an Ar gas flux and become linearly aligned in the presence of a homogeneous magnetic
field. Sintering promotes rigid nanorod formation. Oleic acid is evaporated as a surfactant
to avoid agglomeration between as-grown nanorods. Further downstream the oleic-acid-
coated nanorods are collected on a cold finger held at 77 K.
pump in order to establish a continuous Argon gas flow of 40 sccm read-
ing a pressure of 20 mbar. Elemental iron was heated and evaporated from
a tungsten crucible into the Argon gas flux using an induction coil operated at
a frequency of 700 kHz and a maximum power output of 20 kW. Supersatura-
tion of Fe in the Ar gas leads to nucleation of Fe clusters. Subsequent growth by
condensation and coagulation results in Fe-nanoparticles which further down-
stream tend to form, depending on their local density in the gas phase, particle
aggregates – dimers, trimers, etc. Since the evaporator was located in the ho-
mogeneous field regime at the entrance of a Halbach-type permanent magnet,
linear aggregation of individual nanoparticles becomes strongly favored. First,
due to the magnetic torque experienced by all nanoparticles, which make their
magnetic moments rotate parallel to the field direction. Whenever the magnetic
torque couples sufficiently strong to an easy axis and/or the shape of the par-
ticles, it impels a rotation of these particles themselves. Secondly, owing to
the dipolar interaction which favors a head-to-tail arrangement of particulate
dipoles among all other possible configurations, the emergence of linear par-
ticle aggregates seems highly reasonable. The enhanced temperature extending
along the Halbach magnet enables sintering [18] of these anisometric particle
aggregates along their way downstream. It is the reduction of the particles free
surface energy that yields the driving force for sintering, which causes neck
formation between neighboring particles resulting in rod-like magnetic objects
with solid-like rigidity, concisely called nanorods.
In order to prevent agglomeration of already grown nanorods in the Ar gas,
we coated individual nanorods by evaporation of oleic acid using a thermally
heated quartz glass crucible placed at the exit of the Halbach magnet. Varia-
tion of the distance between the rf coil and the surfactant source allowed us
to roughly control the rod length: a shorter rod length appeared when decreas-
ing this distance. The aerosol jet exiting the Halbach magnet is directed toward
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248
R. Birringer et al.
a LN2 cooled cold finger where the oleic acid coated nanorods become de-
posited due to thermophoresis. After turning off the evaporation sources, the
Ar gas flux was still maintained until the cold finger temperature approached
room temperature. Then the Ar gas pressure was increased to ambient pressure
and via a leak valve Ar gas was slowly exchanged by air in order to passivate
the surface of the oleic acid coated nanorods by a thin oxide layer thus avoiding
full oxidation of available iron and rendering the samples appropriate for hand-
ling under atmospheric conditions. The soot-like material collected on the cold
finger was rinsed off using heptane and accumulated in a container. Adding
a polar solvent (acetone) makes the oleic-acid-coated particles to agglomer-
ate, which allows then to separate them from the solvent by using a permanent
magnet. Repeated redispersing of the coated particles in heptane and separa-
tion eventually leads to a stable FF. The volume concentration of the magnetic
phase immersed in the carrier fluid was determined by magnetometry yielding
the magnetic moment which is directly proportional to the volume of magnetic
objects whenever they become saturated through an applied field. We preferred
to prepare FF with a volume concentration of magnetic phase of 1–2 vol.-‰
to assure that such FF agree with the limiting case of a dilute, non-interacting
system. The structure and morphology of nanorods has been investigated by
means of TEM (Jeol JEM 2010).
As discussed in the theory section, the Brownian relaxation frequency fB,
which enables access to the aspect ratio, is related to the complex ac-
susceptibility χ( f ). In fact, fB coincides with the frequency f correlated to
the maximum value of the imaginary part χꢁꢁ( f ) of χ( f ) = χꢁ( f )+iχꢁꢁ( f ).
Using a Quantum Design PPMS (Model 6000), we measured χ( f, T ) and also
obtained χꢁꢁ( f, T ) by means of sinusoidally varying the applied magnetic field
between 10 Oe at frequencies covering the range between 10–104 Hz.
The magnetoviscous behavior of nanorod FF has been investigated by uti-
lizing a PAV (piezo-axial-vibrator) rheometer [51]. The specimen chamber
has a disc-shaped geometry of 20 mm diameter, a height of 5 µm and can be
operated under closed-system conditions. The bottom part of the chamber is
designed to act as a piezo-driven membrane vibrating at frequency f and an
amplitude of 5 nm in this way forcing the nanorod FF to sustain dynamical
squeeze-flow. The frequency-dependent viscosity is deduced from the complex
spring constant of the vibrating system [52]. Using a solenoid oriented with
its axis perpendicular to the driving membrane, a magnetic dc-field of variable
strength can be applied to the specimen chamber.
4.3 Results and discussion
In Fig. 8 we show a representative bright field micrograph of an assemblage of
nanorods prepared by the magnetic-field-assisted aerosol process discussed in
the previous paragraph. A statistical evaluation of the geometry of objects was,
in general, hampered by the circumstance that they exhibited a strong tendency
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Magnetic Nanorods: Genesis, Self-Organization and Applications
249
Fig. 8. Morphology of nanorods. Representative TEM bright field micrographs of aerosol-
grown nanorods at low (a) and high (b) magnifications.
Fig. 9. Neck formation and iron-oxide core shell structure. (a) TEM dark field image dis-
playing intergrain neck formation. (b) TEM dark field micrograph revealing the existence
of a thin oxide layer, marked by arrows, surrounding the Fe-core of grain 2.
to agglomerate during preparation of the TEM specimen, so causing projection
artefacts which in turn gave rise to high uncertainty of the values determined
for their individual length. This problem is due to attractive magnetic dipo-
lar forces between individual rods which tend to impel agglomeration as the
solvent evaporates. As a consequence, we restrict our analysis to estimating
the overall diameter of nanorods; an effective aspect ratio of nanorods will be
obtained from the ac-susceptibility as shown in the next paragraph. TEM mi-
crographs displayed in Fig. 9 reveal clear evidence for neck formation between
individual particles – so providing the nanorods with rigidity – and the for-
mation of a thin oxide layer protecting the core volume of particles/nanorods
from further oxidation. The ac-susceptibility measurements allowed to deduce
the imaginary part χꢁꢁ( f ) of the complex susceptibility of a nanorod FF. The
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250
R. Birringer et al.
Fig. 10. Ac-susceptibility of a nanorod FF. Imaginary part of the susceptibility of a nano-
rod FF at 4 different temperatures and Hdc = 0. The different lines represent fits to the
data points using Eq. (20). The data set taken at T = 150 K displays the signal of a frozen
fluid, indicating that Brownian relaxation is missing. Brownian relaxation frequencies can
be deduced from the maxima of the upper three curves.
experimental data shown in Fig. 10 were analyzed in terms of the Cole–Cole
model [53]
ꢇ
ꢁ
ꢂ
πβ
sin
χ0 −χ
∞
χꢁꢁ(2π f ) =
,
(20)
2
ꢁ
ꢂ
πβ
2
cosh (β ln (2π fτCC))+cos
2
which could fit all data points and yielded the frequencies fB associated with
the maximum values of χꢁꢁ( f ). The Cole–Cole model represents an empirical
modification of the Debye-relaxation. The parameter β has been introduced as
a measure of the width of a size distribution of individual objects exposed to
the ac-field and τCC denotes the average relaxation time. According to Eq. (19),
determination of the effective aspect ratio, neff, based on experimental data for
fB(n) relies upon a reasonable approximation to the geometry of nanorods. For
the sake of simplicity, we assume a cylinder geometry so implying Vhnyrdod(n) ≈
Vhcyydl(n). The hydrodynamic nanorod volume is modeled by
nrod
hyd
V
≈ (π(r +t)2)(n2r +2t) ,
(21)
where t is a measure of the thickness of the surfactant layer (oleic acid) bonded
to the solid core-structure of the nanorods, r is the rod diameter and n is the
aspect ratio representing the number of spheres of diameter 2r that would fit
into the hollow cylinder with an inner radius of r and an outer radius of (r +
t). Concerning further data analysis, we fixed r ꢂ 5 nm based on TEM re-
sults, and supposed t is reasonably approximated by a monolayer of oleic acid
having a thickness of 1–1.5 nm. We note that the data shown in Fig. 10 sug-
gest that the onset of Brownian relaxation should correlate with the melting
transition of the carrier fluid; below the melting point, Brownian relaxation
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Magnetic Nanorods: Genesis, Self-Organization and Applications
251
Fig. 11. Aspect ratio deduced from the temperature-dependent ac-susceptibility. Pseudo-
3d chart of the imaginary susceptibility χꢁꢁ( f, T ) of a nanorod-FF at Hdc = 0 [see Fig. 10].
Black dots (•) indicate maximum values of χꢁꢁ characteristic of Brownian relaxation fre-
quencies. Full lines show computed relaxation frequencies for different aspect ratios n and
a nanorod diameter d = 11 nm, representing an average diameter obtained from TEM for
the investigated nanorod-FF. The vertical line at T = 183 K indicates the melting point of
the carrier fluid heptane.
should cease. Any signal in this regime would originate from superparamag-
netic (Neelian) relaxation [54]. In order to verify this conjecture, we measured
frequency- and temperature-dependent ac-susceptibilities to obtain χꢁꢁ( f, T )
as shown in Fig. 11. It is straightforward now to analyze the maximum of
χꢁꢁ( f, T ) in terms of fB(n, T ) where n is treated as a parameter, implying that
Vhsyd in Ds is substituted by Vhnyrdod(n). The temperature dependence of fB reveals
the temperature dependence of the rotational diffusion coefficient. The Brown-
ian relaxation frequencies derived from the Cole–Cole analysis are depicted as
black dots. The full lines are fits to the data points using Eq. (19). Superpara-
magnetic relaxation below TM seems absent. For integer values of n, it is shown
that n = 24 is a fairly good approximation to the data points. In this context n is
characteristic of an effective aspect ratio, neff , ensemble-averaged over an a pri-
ori unknown distribution of geometrical/morphological degrees of freedom of
nanorods probed by χꢁꢁ( f, T ). We found that n does not depend on temperature
and frequency. What is more, we also verified that n is independent of the mag-
nitude of a superimposed magnetic dc-field. Dc-bias-fields were successively
applied up to a magnitude of 20 mT and, as suggested by Waldron et al. [55],
we observed for all fields applied only a shift of the relaxation frequency, how-
ever, no concomitant change of n. The fairly large value of n seems to indicate
that among all others the largest nanorods may control the overall behavior of
the complex fluid. We conclude by pointing out that all observations made so
far imply that nanorods manifest a pronounced aspect ratio in conjunction with
solid-like rigidity.
The effect of shape anisotropy on the magnetoviscous behavior of nanorod-
FF has been investigated by utilizing a PAV (piezo-axial-vibrator) rheome-
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252
R. Birringer et al.
Fig. 12. Magnetoviscous effect of a nanorod FF. (a) Magnetic field dependent viscosity
change as obtained from a PAV viscosimeter at different frequencies. Lines represent least
squares fits to the experimental data based on Eq. (15). The volume concentration of the
magnetic phase corresponds to φ = 0.9 vol.-‰. (b) Frequency-dependent viscosity change
for different magnetic fields. Full lines reflect least squares fits based on Eq. (20). The shift
of maxima to higher frequencies with increasing field strength indicates a deviation from
pure orientational relaxation behavior.
ter [51]. In Fig. 12 we display the relative change of viscosity of a nanorod-FF
as a function of applied field and frequency, respectively. With the volume frac-
tion φ of the magnetic phase determined by magnetometry to be 0.9 vol.-‰ –
assuming that all magnetic material is made up of Fe – we analyzed the data
points shown in Fig. 12a in terms of Eq. (15). The full lines depict a least
squares fit with τ being the central fit parameter. Fig. 12b reveals a magnetic
⊥
field-dependent shift of maxima of ∆η/η0 to higher frequencies with increas-
ing field strength; the fit to the data points is based on the Cole–Cole formula,
Eq. (20). This shift supports the idea that, apart from mere orientational re-
laxation behavior of the nanorods, resonance behavior becomes superimposed
with the applied homogeneous magnetic field acting as the restoring torque.
Finally, we compare the magnetoviscous effect of nanorod FF with conven-
tional commercially available FF. In order to make this comparison convincing,
we normalize the relative change of viscosity to the volume fraction φ of the
magnetic phase immersed in the carrier fluid. In Fig. 13 the quantity ∆η/φη0
for a conventional FF, investigated by Odenbach and coworkers, is contrasted
with the nanorod FF. A direct comparison of the two sets of data would re-
quire a relation between the shear-rate- and frequency-dependent viscosity.
Although there is no general theory describing the relation between η(γ) and
˙
η( f ), for simple liquids the Cox–Merz theorem postulates in its rigorous ver-
*
*
sion η(γ) = |η ( f )|, where η ( f ) is the complex viscosity; for complex fluids
˙
this identity brakes down. Recently, Chae et al. [56] investigated the valid-
ity of the Cox–Merz rule for magnetic dispersions. They could demonstrate
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253
Fig. 13. Shear-rate- and frequency-dependent magnetoviscous effect of a conventional and
a nanorod-FF. Full black symbols denote the conventional, highly concentrated FF and
open symbols characterize the nanorod-FF. The relative viscosity change is normalized to
the volume fraction φ of magnetic phase. Frequency values can be translated into shear
rates through division by 10 for direct comparison.
that the slopes of the viscosity and the complex viscosity as a function of γ
˙
or f , respectively, are nearly identical. When they shifted the frequency by
a slightly concentration dependent but otherwise constant factor a ≈ 0.1, they
*
found the relation η(γ) ≈ |η (a f )| reasonably good obeyed. Therefore, by di-
˙
viding the frequencies given in Fig. 13 by a factor of 10, the viscosity data
of the nanorod FF and the conventional FF are expected to be comparable
on a one to one basis. A certainly striking feature is that ∆η/φη0 of conven-
tional FF exhibits a decrease over two orders of magnitude with increasing
shear rate, whereas, only little change was detected for nanorod FF. In detail,
at a shear rate of 10−1 s−1 and raising field strength, ∆η/φη0 of conventional
FF reveals a restrained increase (concave-up), in contrary, the nanorod FF is
characterized by a pronounced enhancement up to virtually saturation behav-
ior within less than 10 mT of applied field. Such a behavior is in agreement
with the idea that in conventional FF, with the given high volume fraction
φ ꢀ 0.1 of magnetic phase, raising field strength triggers chain formation of
dipolar interacting nanoparticles, and growing chain length goes along with an
increase in ∆η/φη0 [40]. On the other hand, the solid-like rigidity of nanorods,
which are present in the carrier fluid also at zero field, makes this complex
fluid susceptible to small applied fields (B ꢁ 10 mT ) and as a consequence
results in a concomitant severe change of ∆η/φη0. In the shear rate or fre-
quency regime which is of technical relevance (γ, f ꢀ 10 s−1), ∆η/φ η of the
˙
0
conventional FF drops by two orders of magnitude, in contrast to the nanorod-
FF which yields evidence for, on that scale, a nearly frequency-independent
behavior. As conjectured, the solid-like rigidity of nanorods renders them resis-
tant against mechanical forces/torque mediated by the fluid phase. By contrast,
the applied-field-enhanced build-up of dipolar coupled chain-like aggregates in
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254
R. Birringer et al.
conventional FF results in shear unstable objects, which become basically dis-
integrated into their individual building blocks whenever the shear rate exceeds
10 s−1.
In summary, we succeeded in preparing a nanorod-FF, a complex fluid
made up of shape-anisotropic magnetic particles – called nanorods – immersed
in a nonmagnetic carrier fluid. These anisometric permanent dipoles have
solid-like rigidity and behave as Brownian particles. Nanorod-FF manifest a gi-
ant magnetoviscous effect, particularly, in the low field regime (ꢁ 10 mT) and
at measured frequencies/shear rates up to 103/102 s−1. Therefore, nanorod-
FF recommend themselves as a novel class of complex fluids with a major
potential for technical applications. Since the magnetoviscous effect can be
“switched on” at low magnetic fields as well as in the regime of technical rel-
evant shear rates, applications in the areas of microfluidics and MEMS and/or
NEMS appear plausible.
Acknowledgement
We are grateful to P. Pechold (Ulm University) for having supported us with the
PAV measurements. We thank D. Junk for having made TEM results available
to us.
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